A274778 Number of proper mergings of an n-antichain and an n-chain.

0, 3, 26, 442, 12899, 582381, 37700452, 3315996468, 380835212037, 55380159334315, 9950025870043126, 2165134468142294430, 561245519520167902471, 170913803045738754172185, 60421582956702701927410120, 24543570079301728283314502248, 11353373604627607560431407875081

Offset: 0

Author

Henri Mühle, Nov 11 2016

Keywords

• nonn

Comments

a(n) is also the number of monotone (n+1)-colorings of a complete bipartite digraph K(n,n), where a monotone (n+1)-coloring is a labeling w of the vertices of K(n,n) with integers in {1,2,...,n+1} such that for every arc (e1, e2) we have w(e1) <= w(e2).

Examples

			For n=1, the three proper mergings of a 1-chain {x} and a 1-antichain {y} are x<y, y<x, and x,y.
		

Links

• H. Mühle, Counting Proper Mergings of Chains and Antichains, Discrete Math., Vol. 327(C), 2014, 118-129. Also arXiv:1206.3922 [math.CO], 2012.

Crossrefs

Cf. A085465.

Programs

Maple

a := n -> add(((n-i+1)^n-(n-i)^n)*(i+1)^n, i=0..n):
seq(a(n), n=0..16); # Peter Luschny, Nov 11 2016

Mathematica

a[0] = 0; a[n_] := Sum[((n-i+1)^n - (n-i)^n)*(i+1)^n, {i, 0, n}];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jul 14 2018, after Peter Luschny *)

PARI

a(n) = sum(i=1, n+1, ((n+2-i)^n - (n+1-i)^n)*i^n); \\ Michel Marcus, Jul 14 2018

Formula

a(n) = Sum_{i=1..n+1} ((n+2-i)^n - (n+1-i)^n)*i^n.